How is that energy actually being transferred to the Moon? The creation of tides requires energy, so I would expect that this should take energy from the Moon, slow it and cause it to eventually fall back to the Earth. Why isn't that happening?

Finally, if this is the general mechanism, would other moons that orbit around planets with a liquid surface and causing tides, be receding from their parent planets?

Note the recession only happens when the primary is spinning faster than the satellite. When the satellite is moving faster than the primary (like Phobos and Mars), then it spirals in, not out.
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BowlOfRedJan 21 at 3:43

@РСТȢѸФХѾЦЧШЩЪЫЬ: Um, "closer to the Sun"? It also gets further from the Sun, half the time...
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ThePopMachineJan 22 at 19:36

2 Answers
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The Moon creates tides. Due to tides, the water bulges out towards the Moon (and also on the opposite side).

But the Earth also rotates pretty fast (once a day), faster than the Moon orbits the Earth (once a month). There's friction between the rotating Earth, and the watery bulge created by tides. The rotation of the Earth "wants" to rotate the bulge faster.

In effect, the rotation of the Earth drags the tidal bulge forward - the bulge is always a bit ahead of the Moon. When the Moon is at meridian, the tide is already decreasing.

So there's a bit of extra watery mass on Earth, a little bit ahead of the Moon. This watery bulge interacts gravitationally with the Moon.

This has two effects:

it slows down the rotation of the Earth, gradually sucking energy from it (the Moon pulls the bulge, and therefore the Earth, "back")

that energy is dumped into the Moon's orbital motion, effectively "pulling" it forward

When you dump energy of motion into an orbiting body, it settles into a higher orbit - higher orbit means more energy. Therefore, the transfer of energy from Earth' spin to Moon's orbit gradually makes Moon's orbit larger and larger.

This only happens because the Earth is spinning faster than the Moon orbits it. If the Earth was tidally locked to the Moon (spinning exactly as fast as the Moon orbits it), then no transfer would happen. If the Earth was spinning slower than the Moon's orbit, then the transfer would be opposite (from Moon's orbital motion to Earth's spin).

Note: Counterintuitively, a satellite with more energy actually moves slower, but in a higher orbit. The extra energy goes into raising the orbit, not into making its speed faster. Why this happens exactly is a whole 'nother discussion.

You have correctly identified that the tidal forces are transferring energy from the Earth to the Moon. This energy causes the Moon's orbit to get larger thus slowing it down.

It's a bit counter intuitive, but if you think about it the Earth spins a rate of 1 spin per day The Moon is orbiting the Earth with a period of approximately 27.3 days. If it were to speed up it's orbit would actually decrease bringing it closer to the Earth.

To answer your final point all other moons cause tides on their parent planets and are moving away from them, but the effects are much smaller due to the larger difference in sizes. The Earth/Moon system is unique in the Solar System as the ratio of the sizes of the bodies are relatively close to each other.

The other planet/moon systems do also have this property though. The moons slowly spin outwards.
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Rory AlsopSep 30 '13 at 19:10

I still don't get how speeding up the Moon would decrease its orbit, from what I remember, the faster the body moves, the more distant is the orbit...
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РСТȢѸФХѾЦЧШЩЪЫЬѢѤЮѦѪѨѬѠѺѮѰѲѴSep 30 '13 at 19:30

@ŁukaszLech That's a misconception, the square of the orbital period is equal to the cube of its average distance (Kepler's 3rd law), but as the size of the orbit only scales linearly with distance, the velocity scales as r^(-1/2), i.e. decreasing with distance.
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GuillochonSep 30 '13 at 19:43

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Examples: At 150 miles up, orbital speed is 17000mph. At 22000 miles up it is only 7000mph.
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Rory AlsopSep 30 '13 at 19:52

Interesting. But when the body is slowing, the centrifugal force reduces, therefore is no longer able to match gravity, and the body falls? This is what happens to low orbit satelites?
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РСТȢѸФХѾЦЧШЩЪЫЬѢѤЮѦѪѨѬѠѺѮѰѲѴSep 30 '13 at 20:55